Topological Degree Method for a Coupled System of $\psi$-fractional Semilinear Differential Equations with non Local Conditions

Year 2024, Volume: 7 Issue: 3, 157 - 167, 29.09.2024

Baıhı Asmaa , Ahmed Kajounı , Khalid Hilal , Lmou Hamid

https://doi.org/10.33434/cams.1442676

Abstract

This paper explores the existence of solutions for non-local coupled semi-linear differential equations involving $\psi$-Caputo differential derivatives for an arbitrary $l\in (0,1)$. We use topological degree theory to condense maps and establish the existence of solutions. This theory allows us to relax the criteria of strong compactness, making it applicable to semilinear equations, which is uncommon. Additionally, we provide an example to demonstrate the practical application of our theoretical result.

Keywords

$\psi$-Caputo differential derivatives, Coupled semilinear differential equations, Topological degree method

References

• [1] S. Asawasamrit, A. Kijjathanakorn, S. K. Ntouya, J. Tariboon, Nonlocal boundary value problems for Hilfer fractional differential equations, B. Korean Math. Soc., 55 (2018), 1639-1657.
• [2] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer: New York, NY, USA, 2010.
• [3] L. Gaul, P. Klein, S. Kemple, Damping description involving fractional operators, Mech. Syst. Signal Process, 5 (1991) 81-88.
• [4] H. Lmou, K. Hilal, A. Kajouni, Topological degree method for a y-Hilfer fractional differential equation involving two different fractional orders, J. Math. Sci., 280 (2024), 212–223. https://doi.org/10.1007/s10958-023-06809-z
• [5] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637.
• [6] R. A. Khan, K. Shah, Existence and uniqueness of solutions to fractional order multipoint boundary value problems, Commun. Appl. Anal., 19 (2015), 515–526.
• [7] H. Lmou, K. Hilal, A. Kajouni, A new result for y-Hilfer fractional Pantograph-type Langevin equation and inclusions, J. Math., 2022, Article number: 2441628.
• [8] Z. H. Liu, J. H. Sun, Nonlinear boundary value problems of fractional differential systems, Comp. Math. Appl. 64 (2012), 463-475.
• [9] F. Mainardi, Fractional Diffusive Waves in Viscoelastic Solids, In: J. L. Wegner, F. R. Norwood (Eds.), Nonlinear Waves in Solids, ASME Book No. AMR 137, Fairfield, (1995), 93-97.
• [10] F. Mainardi, P. Paradis, R. Gorenflo, Probability distributions generated by fractional diffusion equations, In: J. Kertesz, I. Kondor (Eds.), Econophysics: An Emerging Science. Kluwer Academic, Dordrecht, (2000).
• [11] M.B. Zada, K. Shah, R.A. Khan, Existence theory to a coupled system of higher order fractional hybrid differential equations by topological degree theory, Int. J. Comput. Appl. Math., 4 (2018), Article number: 102.
• [12] R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A, 278 (2000), 107-125.
• [13] J.V.D.C. Sousa, E.C. Capelas de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of y-Hilfer operator, Differ. Equ. Appl., 11 (2019), 87-106.
• [14] A. Suechoei, P.S. Ngiamsunthorn, Existence uniqueness and stability of mild solutions for semilinear y-Caputo fractional evolution equations, Adv. Differential Equations. 2020, 2020, Article number: 114.
• [15] R. Almeida, A Caputo fractional derivative of a function concerning another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017) 460–481.
• [16] R. Almeida, A.B. Malinowska, M.T.T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and its applications, Math. Methods Appl. Sci. 41 (2018), 336–352.
• [17] R. Almeida, M. Jleli, B.Samet, A numerical study of fractional relaxation-oscillation equations involving y- Caputo fractional derivative, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat, 113(3) (2019), 1873–1891. https://doi.org/10.1007/s13398-018-0590-0
• [18] H. Lmou, K. Hilal, A. Kajouni, Topological degree method for a class of y-Caputo fractional differential Langevin equation, Kragujevac J. Math., 50(2) (2026), 231–243.
• [19] U. Riaz, A. Zada, Analysis of (a;b)-order coupled implicit Caputo fractional differential equations using topological degree method, Int. J. Nonlinear Sci. Numer. Simul., 22(7–8) (2021), 897–915.
• [20] M. Iqbal, Y. Li, K. Shah, R. Ali Khan, Application of topological degree method for solutions of coupled systems of multipoint boundary value problems of fractional order hybrid differential equations, Complexity, Hindawi, 2017, 1-9, Article number: 767814. https://doi.org/10.1155/2017/7676814
• [21] J.R. Graef, J. Henderson, A. Ouahab, Topological Methods for Differential Equations and Inclusions (1st ed.), CRC Press, 2018. https://doi.org/10.1201/9780429446740.
• [22] P. Benevieri, A brief introduction to topological degree theory, Curso de MAT 554 Panorama em Matematica: Aulas dos dias 15 e 17 de outubro de 2018. https://www.ime.usp.br/ pluigi/lezioni-15e17-ott-18
• [23] W.V. Petryshyn, Generalized topological degree, and semilinear equations, Bull. Amer. Math. Soc., 34(2) (1997), 197-201. S0273-0979(97)00716-7.
• [24] K. Muthuselvan, B. Sundaravadivoo, S. Alsaeed, K.S. Nisar, A new interpretation of the topological degree method of Hilfer fractional neutral functional integro-differential equation with nonlocal condition, AIMS Math., 8(7) (2023), 17154-17170. https://www.aimspress.com/aimspress-data/math/2023/7/PDF/math-08-07-876.pdf
• [25] H. Yang, Existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup, Abstr. Appl. Anal., (2012) (SI01) 1-15. https://doi.org/10.1155/2012/903518
• [26] S. Zorlu, A.Gudaimat, Approximate controllability of fractional evolution equations with y-Caputo derivative, Symmetry, 15(5) (2023), 1050. https://doi.org/10.3390/sym15051050.
• [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.